Symplectic Structures and Geometry of Canonical Bundle

我们把我们的研究集中在“流形的Hitchin-Kobayashi对应”上，这是我们项目中最有趣的话题之一。对于一个Fano流形，这样的对应关系被认为是将流形的Chow-Mumford稳定性与Kahler-Einstein度量的存在性联系起来。作为第一步，我们得到：（1）对于Kahler-Einstein度量的一个推广，我们证明了在给定的Fano流形上这类度量模全纯自同构的唯一性。因此，即使在这种广义的上下文中，上述对应关系也是一对一的。(In对于Kahler-Ricci孤子，Tian和Zhu也得到了类似的结果。（2）在与H.Nakagawa的合作中，我们成功地刻画了Fano流形的Futaki特征标，它是流形的Chow-Mumford半稳定性的障碍。(3)在极化流形的复分析研究中，流形的渐近稳定性通过Bergman度量的渐近行为与极化Kahler类中常数量曲率度量的存在性有很强的关系。

We focussed our study on the "Hitchin-Kobayashi correspondence for manifolds" which is one of the most interesting topics in our project. For a Fano manifold, such a correspondence is supposed to relate the Chow-Mumford stability of the manifold with the existence of Kahler-Einstein metrics. As a first step, we obtained :(1) For a certain generalization of Kahler-Einstein metrics, we showed the uniqueness of such metrics modulo holomorphic automorphisms on a given Fano manifold. Hence, even in this generalized context, the above correspondence is one-to-one. (In the case of Kahler-Ricci solitons, a similar result was obtained also by Tian and Zhu.)Recently, we also saw the following :(2) In a joint work with H.Nakagawa, we succeeded in characterizing the Futaki character of a Fano manifold as an obstruction to Chow-Mumford semistability of the manifold.(3) In complex analytic studies of a polarized manifold, the asymptotic stability of the manifold has a strong relationship, via the asymptotic behavior of the Bergman metrics, with the existence of metrics of constant scalar curvature in the polarized Kahler class.